If we set \(b=0\) , the Hamiltonian of the second-order DSM will reduce to the one of the first-order DSM.
This corresponds to the limit \(T_c\rightarrow +\infty \)  or \(L_c\rightarrow 0\) . Our results become those
for the first-order DSM in this limit. On the other hand, if we consider the highly anisotropic limit,
i.e., \(v_{\perp }=0\) , the Hamiltonian of the second-order DSM will be similar to that of a double WSM,
except that the \(2\times 2\)  Pauli matrices in the double WSM are replaced the \(4\times 4\)  \(\Gamma \) 
matrices in the present case. This corresponds to the limit \(T_c\rightarrow 0\)  or \(L_c\rightarrow +\infty \) .
In fact, the functional form of the vacuum polarization in this limit is identical to that for a double
WSM[1]}, [2]}.
